As a piece of university-bashing, it’s hardly on the level of Gordon Brown’s crude attacks on Oxford during the notorious Laura Spence affair, after Labour’s introduction of tuition fees. Still, there’s profound chutzpah in government criticising elite universities for being too exclusive when ministers’ own funding policies push them relentlessly in that direction.

So it is that a joint report from the Office of Fair Access (Offa) and the Higher Education Funding Council chides the most selective universities for making little or no progress in improving access for students from disadvantaged backgrounds.

Offa director Professor Les Ebdon does concede that this lack of progress is “despite considerable efforts” by those institutions. Indeed: universities in the elite Russell Group are bending over backwards to attract poorer students via outreach programmes, bursaries and scholarships.

But you don’t need one of their degrees in rocket science to see that tuition fees of up to £9,000 a year might be part of the problem. The average for all undergraduates starting in England this autumn will be more than £8,500 a year; a third of institutions will charge the full £9,000.

Fees that size, in place since last autumn, were ostensibly a money-saving measure, supposed to save £1.17 billion. That assumption itself looks very doubtful. Earlier this year a report from universities think tank million+ suggested that the economic cost of fees could end up being six-and-a-half times the Treasury’s projected savings. The Treasury has admitted that it will write off almost 40 per cent of fees loans as unrecoverable.

It’s hard to avoid the impression that ministers also brought in fees, scrapping the old block-grant system of funding for most teaching, simply because they wanted to be shot of the universities and their quarrelsome professors.

But most of all, it was always obvious that jacking up fees would make it even more daunting for disadvantaged applicants trying to get into the top institutions. Proponents of the new system are quick to point to the funds available for the poorest students — Offa’s report says that 35,000 received help from a new national scholarship fund. But these ivory towers are already forbidding enough to a 17-year-old from, say, a tough south London housing estate, without having to negotiate their way through a thicket of funding packages.

It is true that the elite universities, especially Oxbridge, didn’t make enough effort to seek out bright working-class students in the past — though I have to say that when I attended Cambridge in the 1980s it was more socially inclusive than the elite (and very expensive) US university where I went for my PhD, a paragon of the kind of market system ministers would like to see here. That has changed radically — but now elite English institutions are having to swim against the tide more than ever.

Our universities have a crucial role in improving this country’s lamentable levels of social mobility. But they can’t do it on their own: most of the task of giving bright working-class kids a chance to move up has to happen elsewhere in the education system, before they reach 18. And saddling a generation of young people with huge debts just makes it even harder. If, that is, this well-educated Cabinet really cares about social mobility at all.